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Understanding the domain of a function is crucial when picturing how a function behaves. The domain reflects all the possible input values that a function can accept without causing any complications like division by zero or square roots of negative numbers. For example, the functions \(f(x) = \sqrt{x}+1\) and \(g(x) = \sqrt{x+1}\) have domains that reflect the limitation of the square root operation.

The square root demands non-negative inputs, consequently, both functions have a domain of [0, ∞). It is important to start by establishing such boundaries because it informs where on the x-axis the graph will exist. When tasked with sketching these functions, first check for operations within the function that impose restrictions on the domain, mark these on your graph, and know that your function's graph will not exist outside of these marks.

The range of a function, complementing the domain, signifies all the potential output values the function can produce. In case of square root functions like our \(f(x) = \sqrt{x}+1\) and \(g(x) = \sqrt{x+1}\), we determine the range by evaluating the smallest value they can take and observing their behavior. Since square roots output non-negative values, the smallest value for \(f(x)\) when \(x = 0\) is 1, leading to a range of [1, ∞). Similarly, \(g(x)\) takes its smallest value at \(x = 0\), yielding 0, which defines its range as [0, ∞).

When graphing these functions, the range guides us in identifying from which point on the y-axis the graph will start and how it will progress as x increases. It's a valuable indicator of the vertical stretch of the graph.

Graphing functions is the art of portraying the behavior of mathematical expressions visually. The process involves plotting points onto a coordinate plane, affording us a clear representation of how a function acts over its domain. With functions like \(f(x) = \sqrt{x}+1\) and \(g(x) = \sqrt{x+1}\), key plot points include intercepts and understanding the increasing nature of the square root.

When sketching, start by plotting the y-intercept and note that \(f(x)\) lacks an x-intercept due to being always positive. Then, as the square root functions naturally increase, draw the graphs as curves moving up and to the right. Include more points to ensure accuracy, especially within the given interval, in our case [0,4]. Remember, the graphs should reflect the natural growth of the square root functions combined with any vertical shift the addition in the function might introduce.